We are thinking of maybe an izakaya but somewhere, where there is atmosphere but also fun and probably a bit of drinking
If you like music, there is a concert festival in Makuhari Messe called "COWNTDOWN JAPAN"
Many Japanese famous bands (mainly rock) will be on stage and there is a countdown on each stage. I have been there few times and enjoyed very much.
I have been to a countdown event in Tokyo Disney Sea, which was extremely freezing but extremely fun! However, the ticket is limited to pre-sell (and because it is so popular, there is a lottery to purchase the ticket) and I think it is hard to get a ticket from now.
If you visit temples, they might offer visitors to participate in ringing the New Year's Eve bell called "Joya No Kane". Joya No Kane is rang 108 times before we welcome New Year, to think back about the past year and wish happiness for the coming year.
For example Zojoji Temple near Tokyo tower offers visitors to ring the bell, costs 2,000 yen.
Actually, typical and traditional Japanese style to spend New Years Eve is stay at home. Families gather and watch a TV Show called "Kohaku Uta Gassen", which is like a music show popular artists of the year appear and sing songs, then eat Toshikoshi Soba noodles, and welcome new year at home. Some people visit temples or shrines after midnight (called "Hatsumode") after that.
There is a high possibility that the restaurant in the big city is open.
Shibuya, Shinjuku, Roppongi, Ikebukuro
If you walk around the area you might find interesting places.
I found bars that are open in NYE.
Ishinohana: here is famous and good atmosphere.
Maduro or Oak Door (I think restaurant of hotels will open)